See: Math, Binomial theorem, Finite fields
Field with one element, {$F_1$}
The field with one element, {$F_1$}, i a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case.
My impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?
Study
- Find q-analogues for each of the four interpretations of the binomial theorem. See how what they count relates to a finite field of characteristic q. Then see what happens to each of them, and their relationship, when q->1.
- How the Hall algebra and Hall polynomials give rise to the symmetric functions and, in particular, Schur functions, when q goes to 0.
Learn
- Learn what is known about the field with one element.
- Learn the underlying algebraic geometry.
- Learn how the field with one element relates to the Riemann hypothesis.
- Learn about finite fields, especially their combinatorics. Be able to contemplate {$F_{1^n}$}.
- Learn how Weyl groups can be thought of as algebraic groups over the field with one element. The symmetric group has n! elements and the General linear group over a finite field has {$[n!]_q$} elements. Relate this to Schur-Weyl duality.
- Learn about the relationship between the two starting points for homographies and projective spaces. One based on fields and one without fields. Consider that as a relationship q->1.
Investigate
- How do finite fields deal with the issue that Lie algebras deal with: how to link countings?
- Does 1-1=0 in the field with one element?
Readings
Introductory
Overview
- Mapping F1-land: An overview of geometries over the field with one element Preprint. J.L. Pena, O. Lorscheid.
- F_un Mathematics
- Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs.
General theories
- Projective geometry over {$F_1$} and the Gaussian binomial coefficients, Henry Cohn, his research
- New Approach to Arakelov Geometry Nikolai Durov. A theory of generalized rings and schemes.
- New foundations for geometry: Two non-additive languages for arithmetical geometry Shai Haran. This point of view of the general-linear-group suggests also that for the field with one element F we have {$"GL_n(\mathbb{F}_1)"=S_n$}, the symmetric group, which embeds as a common subgroup of all the finite group {$GL_n(\mathbb{F}_p)$}, p prime (or the "field" {$\mathbb{F}\{\pm1\}$}, with {$"GL_n(\mathbb{F}\{\pm1\})"=\{\pm1\}^n \bowtie S_n$}. (The symmetry group for {$B_n$} and {$C_n$}.)
- Quantum {$F_{un}$}: The q=1 Limit of Galois Field Quantum Mechanics, Projective Geometry & the Field With One Element Chang, Lewis, Minic, Takeuchi. Model of passing from quantum to classical mechanics
- (Non) Commutative F-un Geometry Lieven Le Bruyn
- Do the symmetric functions have a function-field analogue? Draft. Darij Grinberg.
- Characteristic one, entropy and the absolute point. Alain Connes, Caterina Consani.
Special aspects
Interpretations of mathematical structures in terms of {$F_1$}.
- A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists!
- The “general linear group” in n dimensions over the field of one element is the symmetric group {$S_n$}.
- A group is a Hopf algebra over the field with one element.
- The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other. Orthogonal group The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension)....
- Clifford algebra Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.
F1-believers base their f-unny intuition on the following two mantras :
- F1 forgets about additive data and retains only multiplicative data.
- F1-objects only acquire flesh when extended to Z (or C).
What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18], deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.
Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489.
Ideas
Cognitive ideas regarding {$F_1$}.
- Limit as q->1
- The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q->1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes.
- I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling God's dance.
- A field relates two groups: an additive group (the level) and a multiplicative group (the metalevel of actions). As regards the action, the zero of the additive group is the negation of action - no action taken, whereas the one of the multiplicative group is the action that has no effect. Therein lies the distinction of the level and the metalevel.
- I think that an affine geometry is not so much distinguished by its not having a zero (a zero or origin can always be defined) but by its not having a one. Perhaps a projective geometry has both a zero and an infinity and so a one is naturally available.
- Zero is not a choice. The field needs to offer another choice.
- There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs. There is none (negation=0).
- Fields are "complete" mathematical structures (having all of the operations) but thus inevitably having a "gap" by which 0 and 1 are distinct. This is the quintessential gap and the prime numbers are likewise gaps in the factorization of numbers.
- P and NP. Field with 1 element - deterministic. Char q - nondeterministic.
- Intersection and union do not have inverses.
- Intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not distributive over multtiplication.
- The anharmonic group (see Cross-ratio) permutes 0,1 and infinity.
Finite fields
- Lyndon words - irreducible polynomials for finite fields
- Duality of q and n in {$GL_n(F_q)$}.
- Multiset of Lyndon words - reducible and irreducible. Homogeneous symmetric functions of eigenvalues.
- Interpolation between homogeneous and elementary - between commutativity and anti-commutativity.
- Lyndon words are like prime numbers.
- Dimension of free Lie algebras = number of Lyndon words of length n
- What would be the q-theory for finite fields for matrix combinatorics?
Dear Harvey,
Thank you for your invitations in your letter below and also earlier, "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html
You mentioned, in my words, that you are looking for an issue that working mathematicians are grappling with where the classical ZFC foundations are not satisfactory or sufficient. Would the "field with one element" be such issue for you?
https://ncatlab.org/nlab/show/field+with+one+element
Jacques Tits raised this issue in 1957 and it has yet to be resolved despite substantial interest, conferences, and long papers. Would that count as a "problem" for the Foundations of Mathematics? It seems that in the history of math it is very easy to simply say "that is not real math" as was the case with the rational numbers, imaginary numbers, infinitesimals, infinite series, etc.
The issue is that there are many instances where a combinatorial interpretation makes sense in terms of a finite field Fq of characteristic q, which is all the more insightful when q=1. For example, the Gaussian binomial coefficients can be interpreted as counting the number of k-dimensional subspaces of an n-dimensional vector space over a finite field Fq. When q=1, then we get the usual binomial coefficients which count the subsets of size k of a set of size n. So this suggests an important way of thinking about sets. However, F1 would be a field with one element, which means that 0=1. But if 0 and 1 are not distinct, then none of the usual properties of a field make sense. Nobody has figured out a convincing interpretation for F1. And yet the concept seems to be pervasive, meaningful and fruitful.
If there was an alternate foundations of mathematics which yielded a helpful, meaningful, fruitful interpretation of F1, would that count in its favor? And if it could do everything that FOM can do, then might it be preferable, at least for some? But especially if that interpretation was shown not to make sense in other FOMs?
I am curious what you and others think.
Andrius